Imagine I'm walking in a tunnel and I stop, seeing that I am three eights of the way in. The tunnel is divided into eight equal length sections.

All of a sudden, I hear the engine noise of a train behind me. Now, the speed of the train is unknown. To survive, I have to run out of the tunnel as quickly as possible.

I have two choices, running forwards or backwards. If I run backwards in the direction of the train, I will be able to get out of the tunnel right at the moment the train comes in the tunnel. If I run forwards in the way the train is coming, the train will exit the tunnel at the same time as I do.

4 Answers
4

Let's say that you can run with a speed $v$, and the train, $u$. Let's also assume that the tunnel has length $8L$, and that the train is originally a distance $x$ away from the entrance of the tunnel. Then we have $$\frac{3L}{v} = \frac{x}{u}$$ and $$\frac{5L}{v} = \frac{x+8L}{u}$$

We therefore have $$\frac{5}{3} \cdot \frac{x}{u} = \frac{x+8L}{u}$$

Solving this, we get $$\frac{2}{3} x = 8L \Rightarrow x = 12L$$

Plugging it back in, we get $$\frac{3L}{v} = \frac{12L}{u} \Rightarrow \frac{u}{v} = 4$$

So Anupams question is right! We can find the speed of the man in terms of the speed of the train.
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EminApr 27 '14 at 16:24

2

@Emin: In that sense, though, it doesn't make sense to find the speed of the man in terms of the speed of the ground. Either way I look at it, that comment seems kind of strange.
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user2357112Apr 27 '14 at 16:28

In the time it takes the train to reach the start of the tunnel, you can travel 3/8 the length of the tunnel. That means if you had instead run away from the train, you would be 3/4 of the way through the tunnel when the train reached the entrance. How fast would you have to go to traverse the remaining 1/4 of the tunnel in the time it takes the train to go through the whole tunnel?

According to the data you are giving, you die if you run. Either way, you leave the tunnel just as the train enters or exits. No time to leave the tracks. Big splash, that's it.

Now some people calculated that you run at 1/4th the speed of the train. That's not quite right. You run either 3/8ths or 5/8ths of the tunnel, obviously at maximum speed. It is inevitable that your speed for 5/8ths of the tunnel will be lower because of exhaustion. If you can run 3/8ths of the tunnel at x meter/second, and 5/8ths of the tunnel at y meter/second, and the length of the tunnel is t meters, then your running time it (3t/8x) vs (5t/8y) seconds. The train therefore takes t/8 (5/y - 3/x) seconds to cross the tunnel, which means 8 / (5/y - 3/x) meter per seconds. The proportion of your speed divided by the speed of the train is (5x/y - 3) / 8 over the shorter distance, and (5 - 3y/x) / 8 over the longer distance. The first is a bit more than 1/4, the latter is a bit less, unless x = y.

Since the train is going quite slow, I'd recommend taking off any lose clothing as quickly as possible, then lying down as flat as possible between the tracks and praying while you let the train drive over you.

Funny seeing someone who has no idea what logic really is posting a logic problem. Sorry
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user11355Apr 27 '14 at 21:37

This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post.
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naslundxApr 27 '14 at 22:12

@naslundx The question is "What is my speed?" to which the answer is zero. As the question stands, until you stop thinking and start running, your speed is zero. If the question was "How fast can I run?" then all this fun math would be useful.
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RiverFogApr 27 '14 at 22:20

2

Being pedantic can be fun at times, but this is not a useful answer.
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Henry SwansonApr 27 '14 at 23:56